Derived Manifolds from Functors of Points
Franz Vogler
Cite this publication as
Franz Vogler, Derived Manifolds from Functors of Points (2013), Logos Verlag, Berlin, ISBN: 9783832591649
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Description / Abstract
In this thesis a functorial approach to the category of derived manifolds is developed. We use a similar approach as Demazure and Gabriel did when they described the category of schemes as a full subcategory of the category of sheaves on the big Zariski site. Their work is further developed leading to the definition of C#-schemes and derived manifolds as certain sheaves on appropriate big sites. The new description of C#-schemes and derived manifolds via functors is compared to the previous approaches via locally ringed spaces given by D. Joyce and D. Spivak. Furthermore, it is proven that both approaches lead to equivalent categories.
Table of content
- BEGINN
- 1 Introduction
- 1.1 Motivation and Accomplishments
- 1.2 Organisation of the Chapters
- 1.3 Background and Notation
- 1.4 Acknowledgements
- 2 Analysis of Model Structures
- 2.1 Objectwise Adjoints
- 2.2 The Global Model Structures
- 2.3 Simplicial Enrichment of Model Categories
- 2.4 Absolute derived Functors
- 2.5 Homotopy Limits as absolute derived Functors
- 2.6 The left Bousfield Localization
- 2.7 Sites and their Morphisms
- 2.8 Homotopy Sheaves
- 3 Smooth Functors as C1-Schemes
- 3.1 Commutative Algebra with C1-Rings
- 3.2 C1-rings and Topology
- 3.3 C1-Schemes from Smooth Functors
- 3.4 The Big Structure Sheaf
- 3.5 C1-ringed Spaces as C1-Schemes
- 3.6 From Smooth Functors to C1-Ringed Spaces
- 4 Derived Manifolds as Functors
- 4.1 The Layout for Smooth Rings
- 4.2 Topology for Smooth Rings
- 4.3 Derived Manifolds from Smooth-Simplicial Functors
- 4.4 The Global Structure Sheaf
- 4.5 Derived Manifolds as Ringed Spaces
- 4.6 The Functor approach to LRS